Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(5,9)(7,10)(11,12,16,17)(13,14,19,22)(15,23,24,18)(20,26,25,21)(27,28,31,34)(29,35,37,32)(30,33,39,38)(36,41,42,40);(3,4,6,8)(5,7,9,10)(12,17)(13,19)(15,24)(21,26)(27,28,31,34)(29,35,37,32)(30,38,39,33)(36,40,42,41);(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,20)(16,19)(17,22)(18,21)(23,26)(24,25)(27,28)(29,32)(30,33)(31,34)(35,37)(36,40)(38,39)(41,42);(11,14)(12,13)(15,21)(16,22)(17,19)(18,20)(23,25)(24,26)(27,29,31,37)(28,32,34,35)(30,36,39,42)(33,40,38,41);(11,15,16,24)(12,18,17,23)(13,20,19,25)(14,21,22,26)(27,30)(28,33)(29,36)(31,39)(32,40)(34,38)(35,41)(37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609995);